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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000983
(load all 4 compositions to match this statistic)
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Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 1 => 1
00 => 01 => 2
01 => 10 => 2
10 => 11 => 1
11 => 11 => 1
000 => 001 => 2
001 => 010 => 3
010 => 101 => 3
011 => 101 => 3
100 => 101 => 3
101 => 110 => 2
110 => 111 => 1
111 => 111 => 1
0000 => 0001 => 2
0001 => 0010 => 3
0010 => 0101 => 4
0011 => 0101 => 4
0100 => 1001 => 2
0101 => 1010 => 4
0110 => 1011 => 3
0111 => 1011 => 3
1000 => 1001 => 2
1001 => 1010 => 4
1010 => 1101 => 3
1011 => 1101 => 3
1100 => 1101 => 3
1101 => 1110 => 2
1110 => 1111 => 1
1111 => 1111 => 1
00000 => 00001 => 2
00001 => 00010 => 3
00010 => 00101 => 4
00011 => 00101 => 4
00100 => 01001 => 3
00101 => 01010 => 5
00110 => 01011 => 4
00111 => 01011 => 4
01000 => 10001 => 2
01001 => 10010 => 3
01010 => 10101 => 5
01011 => 10101 => 5
01100 => 10101 => 5
01101 => 10110 => 3
01110 => 10111 => 3
01111 => 10111 => 3
10000 => 10001 => 2
10001 => 10010 => 3
10010 => 10101 => 5
10011 => 10101 => 5
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000982
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 1 => 1
1 => 1 => 1 => 1 => 1
00 => 01 => 00 => 00 => 2
01 => 10 => 11 => 11 => 2
10 => 11 => 10 => 01 => 1
11 => 11 => 10 => 01 => 1
000 => 001 => 011 => 110 => 2
001 => 010 => 000 => 000 => 3
010 => 101 => 111 => 111 => 3
011 => 101 => 111 => 111 => 3
100 => 101 => 111 => 111 => 3
101 => 110 => 100 => 001 => 2
110 => 111 => 101 => 101 => 1
111 => 111 => 101 => 101 => 1
0000 => 0001 => 0100 => 0010 => 2
0001 => 0010 => 0111 => 1110 => 3
0010 => 0101 => 0000 => 0000 => 4
0011 => 0101 => 0000 => 0000 => 4
0100 => 1001 => 1100 => 0011 => 2
0101 => 1010 => 1111 => 1111 => 4
0110 => 1011 => 1110 => 0111 => 3
0111 => 1011 => 1110 => 0111 => 3
1000 => 1001 => 1100 => 0011 => 2
1001 => 1010 => 1111 => 1111 => 4
1010 => 1101 => 1000 => 0001 => 3
1011 => 1101 => 1000 => 0001 => 3
1100 => 1101 => 1000 => 0001 => 3
1101 => 1110 => 1011 => 1101 => 2
1110 => 1111 => 1010 => 0101 => 1
1111 => 1111 => 1010 => 0101 => 1
00000 => 00001 => 01011 => 11010 => 2
00001 => 00010 => 01000 => 00010 => 3
00010 => 00101 => 01111 => 11110 => 4
00011 => 00101 => 01111 => 11110 => 4
00100 => 01001 => 00011 => 11000 => 3
00101 => 01010 => 00000 => 00000 => 5
00110 => 01011 => 00001 => 10000 => 4
00111 => 01011 => 00001 => 10000 => 4
01000 => 10001 => 11011 => 11011 => 2
01001 => 10010 => 11000 => 00011 => 3
01010 => 10101 => 11111 => 11111 => 5
01011 => 10101 => 11111 => 11111 => 5
01100 => 10101 => 11111 => 11111 => 5
01101 => 10110 => 11100 => 00111 => 3
01110 => 10111 => 11101 => 10111 => 3
01111 => 10111 => 11101 => 10111 => 3
10000 => 10001 => 11011 => 11011 => 2
10001 => 10010 => 11000 => 00011 => 3
10010 => 10101 => 11111 => 11111 => 5
10011 => 10101 => 11111 => 11111 => 5
0000000001 => 0000000010 => 0101010111 => 1110101010 => ? = 3
0001100111 => 0010101011 => 0111111110 => 0111111110 => ? = 8
0001010111 => 0010101011 => 0111111110 => 0111111110 => ? = 8
0001001111 => 0010010111 => 0111000010 => 0100001110 => ? = 4
0000111101 => 0001011110 => 0100001011 => 1101000010 => ? = 4
0000110111 => 0001011011 => 0100001110 => 0111000010 => ? = 4
0000011111 => 0000101111 => 0101111010 => 0101111010 => ? = 4
0000000110 => 0000001011 => 0101011110 => 0111101010 => ? = 4
0000011000 => 0000101001 => ? => ? => ? = 5
0000011110 => 0000101111 => 0101111010 => 0101111010 => ? = 4
0001100000 => 0010100001 => ? => ? => ? = 5
0001100110 => 0010101011 => 0111111110 => 0111111110 => ? = 8
0001001000 => 0010010001 => ? => ? => ? = 3
0001001110 => 0010010111 => 0111000010 => 0100001110 => ? = 4
0001000010 => 0010000101 => ? => ? => ? = 4
0001010100 => 0010101001 => 0111111100 => ? => ? = 7
0000010100 => 0000101001 => ? => ? => ? = 5
0000001110 => 0000010111 => 0101000010 => 0100001010 => ? = 4
0000111110 => 0001011111 => 0100001010 => 0101000010 => ? = 4
0000010110 => 0000101011 => 0101111110 => 0111111010 => ? = 6
0001011110 => 0010101111 => 0111111010 => 0101111110 => ? = 6
0000000101 => 0000001010 => 0101011111 => 1111101010 => ? = 5
0000001101 => 0000010110 => 0101000011 => 1100001010 => ? = 4
0000011101 => 0000101110 => 0101111011 => 1101111010 => ? = 4
0000010101 => 0000101010 => 0101111111 => 1111111010 => ? = 7
0000101101 => 0001010110 => 0100000011 => ? => ? = 6
Description
The length of the longest constant subword.
Matching statistic: St000381
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(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => [1] => 1
1 => 1 => [1] => [1] => 1
00 => 01 => [1,1] => [2] => 2
01 => 10 => [1,1] => [2] => 2
10 => 11 => [2] => [1,1] => 1
11 => 11 => [2] => [1,1] => 1
000 => 001 => [2,1] => [2,1] => 2
001 => 010 => [1,1,1] => [3] => 3
010 => 101 => [1,1,1] => [3] => 3
011 => 101 => [1,1,1] => [3] => 3
100 => 101 => [1,1,1] => [3] => 3
101 => 110 => [2,1] => [2,1] => 2
110 => 111 => [3] => [1,1,1] => 1
111 => 111 => [3] => [1,1,1] => 1
0000 => 0001 => [3,1] => [2,1,1] => 2
0001 => 0010 => [2,1,1] => [3,1] => 3
0010 => 0101 => [1,1,1,1] => [4] => 4
0011 => 0101 => [1,1,1,1] => [4] => 4
0100 => 1001 => [1,2,1] => [2,2] => 2
0101 => 1010 => [1,1,1,1] => [4] => 4
0110 => 1011 => [1,1,2] => [1,3] => 3
0111 => 1011 => [1,1,2] => [1,3] => 3
1000 => 1001 => [1,2,1] => [2,2] => 2
1001 => 1010 => [1,1,1,1] => [4] => 4
1010 => 1101 => [2,1,1] => [3,1] => 3
1011 => 1101 => [2,1,1] => [3,1] => 3
1100 => 1101 => [2,1,1] => [3,1] => 3
1101 => 1110 => [3,1] => [2,1,1] => 2
1110 => 1111 => [4] => [1,1,1,1] => 1
1111 => 1111 => [4] => [1,1,1,1] => 1
00000 => 00001 => [4,1] => [2,1,1,1] => 2
00001 => 00010 => [3,1,1] => [3,1,1] => 3
00010 => 00101 => [2,1,1,1] => [4,1] => 4
00011 => 00101 => [2,1,1,1] => [4,1] => 4
00100 => 01001 => [1,1,2,1] => [2,3] => 3
00101 => 01010 => [1,1,1,1,1] => [5] => 5
00110 => 01011 => [1,1,1,2] => [1,4] => 4
00111 => 01011 => [1,1,1,2] => [1,4] => 4
01000 => 10001 => [1,3,1] => [2,1,2] => 2
01001 => 10010 => [1,2,1,1] => [3,2] => 3
01010 => 10101 => [1,1,1,1,1] => [5] => 5
01011 => 10101 => [1,1,1,1,1] => [5] => 5
01100 => 10101 => [1,1,1,1,1] => [5] => 5
01101 => 10110 => [1,1,2,1] => [2,3] => 3
01110 => 10111 => [1,1,3] => [1,1,3] => 3
01111 => 10111 => [1,1,3] => [1,1,3] => 3
10000 => 10001 => [1,3,1] => [2,1,2] => 2
10001 => 10010 => [1,2,1,1] => [3,2] => 3
10010 => 10101 => [1,1,1,1,1] => [5] => 5
10011 => 10101 => [1,1,1,1,1] => [5] => 5
0000000001 => 0000000010 => [8,1,1] => [3,1,1,1,1,1,1,1] => ? = 3
0001011011 => 0010101101 => ? => ? => ? = 6
0001001111 => 0010010111 => [2,1,2,1,1,3] => [1,1,4,3,1] => ? = 4
0000111101 => 0001011110 => [3,1,1,4,1] => [2,1,1,4,1,1] => ? = 4
0000111011 => 0001011101 => [3,1,1,3,1,1] => [3,1,4,1,1] => ? = 4
0000110111 => 0001011011 => ? => ? => ? = 4
0000101111 => 0001010111 => [3,1,1,1,1,3] => [1,1,6,1,1] => ? = 6
0000011111 => 0000101111 => [4,1,1,4] => [1,1,1,4,1,1,1] => ? = 4
0000000000 => 0000000001 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 2
0000000110 => 0000001011 => [6,1,1,2] => [1,4,1,1,1,1,1] => ? = 4
0000011000 => 0000101001 => ? => ? => ? = 5
0000011110 => 0000101111 => [4,1,1,4] => [1,1,1,4,1,1,1] => ? = 4
0000010010 => 0000100101 => ? => ? => ? = 4
0001100000 => 0010100001 => ? => ? => ? = 5
0001001000 => 0010010001 => ? => ? => ? = 3
0001001110 => 0010010111 => [2,1,2,1,1,3] => [1,1,4,3,1] => ? = 4
0001000010 => 0010000101 => ? => ? => ? = 4
0001011010 => 0010101101 => ? => ? => ? = 6
0001010100 => 0010101001 => ? => ? => ? = 7
0000010100 => 0000101001 => ? => ? => ? = 5
0000000010 => 0000000101 => [7,1,1,1] => [4,1,1,1,1,1,1] => ? = 4
0000001110 => 0000010111 => [5,1,1,3] => [1,1,4,1,1,1,1] => ? = 4
0000111110 => 0001011111 => [3,1,1,5] => [1,1,1,1,4,1,1] => ? = 4
0000000100 => 0000001001 => [6,1,2,1] => [2,3,1,1,1,1,1] => ? = 3
0000001010 => 0000010101 => [5,1,1,1,1,1] => [6,1,1,1,1] => ? = 6
0000010110 => 0000101011 => ? => ? => ? = 6
0000101110 => 0001010111 => [3,1,1,1,1,3] => [1,1,6,1,1] => ? = 6
0001011110 => 0010101111 => ? => ? => ? = 6
0000000101 => 0000001010 => [6,1,1,1,1] => [5,1,1,1,1,1] => ? = 5
0000001101 => 0000010110 => [5,1,1,2,1] => [2,4,1,1,1,1] => ? = 4
0000011101 => 0000101110 => [4,1,1,3,1] => [2,1,4,1,1,1] => ? = 4
0000001001 => 0000010010 => ? => ? => ? = 3
0000010101 => 0000101010 => ? => ? => ? = 7
0000101101 => 0001010110 => ? => ? => ? = 6
Description
The largest part of an integer composition.
Matching statistic: St001235
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(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 67%
Values
0 => 1 => [1] => 1
1 => 1 => [1] => 1
00 => 01 => [1,1] => 2
01 => 10 => [1,1] => 2
10 => 11 => [2] => 1
11 => 11 => [2] => 1
000 => 001 => [2,1] => 2
001 => 010 => [1,1,1] => 3
010 => 101 => [1,1,1] => 3
011 => 101 => [1,1,1] => 3
100 => 101 => [1,1,1] => 3
101 => 110 => [2,1] => 2
110 => 111 => [3] => 1
111 => 111 => [3] => 1
0000 => 0001 => [3,1] => 2
0001 => 0010 => [2,1,1] => 3
0010 => 0101 => [1,1,1,1] => 4
0011 => 0101 => [1,1,1,1] => 4
0100 => 1001 => [1,2,1] => 2
0101 => 1010 => [1,1,1,1] => 4
0110 => 1011 => [1,1,2] => 3
0111 => 1011 => [1,1,2] => 3
1000 => 1001 => [1,2,1] => 2
1001 => 1010 => [1,1,1,1] => 4
1010 => 1101 => [2,1,1] => 3
1011 => 1101 => [2,1,1] => 3
1100 => 1101 => [2,1,1] => 3
1101 => 1110 => [3,1] => 2
1110 => 1111 => [4] => 1
1111 => 1111 => [4] => 1
00000 => 00001 => [4,1] => 2
00001 => 00010 => [3,1,1] => 3
00010 => 00101 => [2,1,1,1] => 4
00011 => 00101 => [2,1,1,1] => 4
00100 => 01001 => [1,1,2,1] => 3
00101 => 01010 => [1,1,1,1,1] => 5
00110 => 01011 => [1,1,1,2] => 4
00111 => 01011 => [1,1,1,2] => 4
01000 => 10001 => [1,3,1] => 2
01001 => 10010 => [1,2,1,1] => 3
01010 => 10101 => [1,1,1,1,1] => 5
01011 => 10101 => [1,1,1,1,1] => 5
01100 => 10101 => [1,1,1,1,1] => 5
01101 => 10110 => [1,1,2,1] => 3
01110 => 10111 => [1,1,3] => 3
01111 => 10111 => [1,1,3] => 3
10000 => 10001 => [1,3,1] => 2
10001 => 10010 => [1,2,1,1] => 3
10010 => 10101 => [1,1,1,1,1] => 5
10011 => 10101 => [1,1,1,1,1] => 5
0000000 => 0000001 => [6,1] => ? = 2
0000001 => 0000010 => [5,1,1] => ? = 3
0000010 => 0000101 => [4,1,1,1] => ? = 4
0000011 => 0000101 => [4,1,1,1] => ? = 4
0000100 => 0001001 => [3,1,2,1] => ? = 3
0000101 => 0001010 => [3,1,1,1,1] => ? = 5
0000110 => 0001011 => [3,1,1,2] => ? = 4
0000111 => 0001011 => [3,1,1,2] => ? = 4
0001000 => 0010001 => [2,1,3,1] => ? = 3
0001001 => 0010010 => [2,1,2,1,1] => ? = 3
0001010 => 0010101 => [2,1,1,1,1,1] => ? = 6
0001011 => 0010101 => [2,1,1,1,1,1] => ? = 6
0001100 => 0010101 => [2,1,1,1,1,1] => ? = 6
0001101 => 0010110 => [2,1,1,2,1] => ? = 4
0001110 => 0010111 => [2,1,1,3] => ? = 4
0001111 => 0010111 => [2,1,1,3] => ? = 4
0010000 => 0100001 => [1,1,4,1] => ? = 3
0010001 => 0100010 => [1,1,3,1,1] => ? = 3
0010010 => 0100101 => [1,1,2,1,1,1] => ? = 4
0010011 => 0100101 => [1,1,2,1,1,1] => ? = 4
0010100 => 0101001 => [1,1,1,1,2,1] => ? = 5
0010101 => 0101010 => [1,1,1,1,1,1,1] => ? = 7
0010110 => 0101011 => [1,1,1,1,1,2] => ? = 6
0010111 => 0101011 => [1,1,1,1,1,2] => ? = 6
0011000 => 0101001 => [1,1,1,1,2,1] => ? = 5
0011001 => 0101010 => [1,1,1,1,1,1,1] => ? = 7
0011010 => 0101101 => [1,1,1,2,1,1] => ? = 4
0011011 => 0101101 => [1,1,1,2,1,1] => ? = 4
0011100 => 0101101 => [1,1,1,2,1,1] => ? = 4
0011101 => 0101110 => [1,1,1,3,1] => ? = 4
0011110 => 0101111 => [1,1,1,4] => ? = 4
0011111 => 0101111 => [1,1,1,4] => ? = 4
0100000 => 1000001 => [1,5,1] => ? = 2
0100001 => 1000010 => [1,4,1,1] => ? = 3
0100010 => 1000101 => [1,3,1,1,1] => ? = 4
0100011 => 1000101 => [1,3,1,1,1] => ? = 4
0100100 => 1001001 => [1,2,1,2,1] => ? = 3
0100101 => 1001010 => [1,2,1,1,1,1] => ? = 5
0100110 => 1001011 => [1,2,1,1,2] => ? = 4
0100111 => 1001011 => [1,2,1,1,2] => ? = 4
0101000 => 1010001 => [1,1,1,3,1] => ? = 4
0101001 => 1010010 => [1,1,1,2,1,1] => ? = 4
0101010 => 1010101 => [1,1,1,1,1,1,1] => ? = 7
0101011 => 1010101 => [1,1,1,1,1,1,1] => ? = 7
0101100 => 1010101 => [1,1,1,1,1,1,1] => ? = 7
0101101 => 1010110 => [1,1,1,1,2,1] => ? = 5
0101110 => 1010111 => [1,1,1,1,3] => ? = 5
0101111 => 1010111 => [1,1,1,1,3] => ? = 5
0110000 => 1010001 => [1,1,1,3,1] => ? = 4
0110001 => 1010010 => [1,1,1,2,1,1] => ? = 4
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001330
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 89%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 89%
Values
0 => 1 => [1] => ([],1)
=> 1
1 => 1 => [1] => ([],1)
=> 1
00 => 01 => [1,1] => ([(0,1)],2)
=> 2
01 => 10 => [1,1] => ([(0,1)],2)
=> 2
10 => 11 => [2] => ([],2)
=> 1
11 => 11 => [2] => ([],2)
=> 1
000 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
010 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
110 => 111 => [3] => ([],3)
=> 1
111 => 111 => [3] => ([],3)
=> 1
0000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
0010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0100 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0110 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
1000 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
1001 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
1010 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
1011 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
1100 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
1110 => 1111 => [4] => ([],4)
=> 1
1111 => 1111 => [4] => ([],4)
=> 1
00000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00001 => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
00010 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
00011 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
00100 => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
00101 => 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
00110 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
00111 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
01000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
01001 => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
01010 => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
01011 => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
01100 => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
01101 => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
01110 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
01111 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
10000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
10001 => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
10010 => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
10011 => 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
10100 => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
10101 => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
10110 => 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
10111 => 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
11000 => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
11001 => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
11010 => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
11011 => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
11100 => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
11101 => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
11110 => 11111 => [5] => ([],5)
=> 1
11111 => 11111 => [5] => ([],5)
=> 1
000000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
000001 => 000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
000010 => 000101 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
000011 => 000101 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
000100 => 001001 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
000101 => 001010 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
000110 => 001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
000111 => 001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
001000 => 010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
001001 => 010010 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
001010 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001011 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001100 => 010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
001101 => 010110 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
001110 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
001111 => 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
010000 => 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
010001 => 100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
010010 => 100101 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
010011 => 100101 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
010100 => 101001 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
010101 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
010110 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
010111 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
011000 => 101001 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
011001 => 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
011010 => 101101 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
011011 => 101101 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
011100 => 101101 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
011101 => 101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
011110 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
011111 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
100000 => 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
100001 => 100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
100010 => 100101 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
100011 => 100101 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
100100 => 101001 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
101000 => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000307
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
0 => 1 => ([(0,1)],2)
=> 1
1 => 1 => ([(0,1)],2)
=> 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
10 => 11 => ([(0,2),(2,1)],3)
=> 1
11 => 11 => ([(0,2),(2,1)],3)
=> 1
000 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
001 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 3
010 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 3
011 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 3
100 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 3
101 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
110 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1
111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
0001 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
0010 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4
0011 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4
0100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2
0101 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4
0110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
0111 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
1000 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2
1001 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4
1010 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
1011 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
1100 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3
1101 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
1110 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
00010 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
00011 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
00100 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3
00101 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
00110 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
00111 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
01000 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
01001 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3
01010 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
01011 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
01100 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
01101 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3
01110 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
01111 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
10000 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2
10001 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3
10010 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
10011 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5
10100 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
10101 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
10110 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3
10111 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3
11000 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
11001 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4
11010 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
11011 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
11100 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
11101 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 3
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 4
000011 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 4
000100 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 3
000101 => 001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 5
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000632
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
0 => 1 => ([(0,1)],2)
=> 0 = 1 - 1
1 => 1 => ([(0,1)],2)
=> 0 = 1 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
11 => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
000 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
001 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
010 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
011 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
100 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
101 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
110 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
0001 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
0010 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
0011 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
0100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 - 1
0101 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
0110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
0111 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
1000 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 - 1
1001 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
1010 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
1011 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
1100 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
1101 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
1110 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
00010 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
00011 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
00100 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3 - 1
00101 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
00110 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
00111 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
01000 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
01001 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3 - 1
01010 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
01011 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
01100 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
01101 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3 - 1
01110 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
01111 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
10000 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 - 1
10001 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3 - 1
10010 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
10011 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 5 - 1
10100 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
10101 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
10110 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3 - 1
10111 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3 - 1
11000 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 - 1
11001 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 4 - 1
11010 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
11011 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
11100 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
11101 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 1
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 3 - 1
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 4 - 1
000011 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 4 - 1
000100 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 3 - 1
000101 => 001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 5 - 1
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
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